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Theorem sxsigon 33185
Description: A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsigon ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜(βˆͺ 𝑆 Γ— βˆͺ 𝑇)))
Proof of Theorem sxsigon
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxsiga 33184 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra)
2 eqid 2732 . . . 4 ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))
3 eqid 2732 . . . 4 βˆͺ 𝑆 = βˆͺ 𝑆
4 eqid 2732 . . . 4 βˆͺ 𝑇 = βˆͺ 𝑇
52, 3, 4txuni2 23068 . . 3 (βˆͺ 𝑆 Γ— βˆͺ 𝑇) = βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))
62sxval 33183 . . . . 5 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
76unieqd 4922 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ βˆͺ (𝑆 Γ—s 𝑇) = βˆͺ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
8 mpoexga 8063 . . . . 5 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
9 rnexg 7894 . . . . 5 ((π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V β†’ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
10 unisg 33136 . . . . 5 (ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)) ∈ V β†’ βˆͺ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) = βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
118, 9, 103syl 18 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ βˆͺ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) = βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
127, 11eqtrd 2772 . . 3 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ βˆͺ (𝑆 Γ—s 𝑇) = βˆͺ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
135, 12eqtr4id 2791 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (βˆͺ 𝑆 Γ— βˆͺ 𝑇) = βˆͺ (𝑆 Γ—s 𝑇))
14 issgon 33116 . 2 ((𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜(βˆͺ 𝑆 Γ— βˆͺ 𝑇)) ↔ ((𝑆 Γ—s 𝑇) ∈ βˆͺ ran sigAlgebra ∧ (βˆͺ 𝑆 Γ— βˆͺ 𝑇) = βˆͺ (𝑆 Γ—s 𝑇)))
151, 13, 14sylanbrc 583 1 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ 𝑇 ∈ βˆͺ ran sigAlgebra) β†’ (𝑆 Γ—s 𝑇) ∈ (sigAlgebraβ€˜(βˆͺ 𝑆 Γ— βˆͺ 𝑇)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βˆͺ cuni 4908   Γ— cxp 5674  ran crn 5677  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  sigAlgebracsiga 33101  sigaGencsigagen 33131   Γ—s csx 33181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-siga 33102  df-sigagen 33132  df-sx 33182
This theorem is referenced by:  sxuni  33186
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